Question: Find the gradient of $f(x, y, z) = zxy - x + \ln(y)$. $\nabla f = ($ $,$ $,$ $)$
Answer: The gradient of a scalar field is all its partial derivatives put together into a vector. For a 3D scalar field, this looks like $\nabla f = (f_x, f_y, f_z)$. Let's find $f_x$, $f_y$, and $f_z$. $\begin{aligned} f_x &= \dfrac{\partial}{\partial x} \left[ zxy - x + \ln(y) \right] \\ \\ &= zy - 1 \\ \\ f_y &= \dfrac{\partial}{\partial y} \left[ zxy - x + \ln(y) \right] \\ \\ &= zx + \dfrac{1}{y} \\ \\ f_z &= \dfrac{\partial}{\partial z} \left[ zxy - x + \ln(y) \right] \\ \\ &= xy \end{aligned}$ The gradient of $f$ is: $\nabla f = \left( zy - 1, zx + \dfrac{1}{y}, xy \right)$